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Research and Development Laboratories
Portland Cement Association
The Flow of Water in Hardened
Portland Cement Paste
T. C. POWERS, H. M. MANN
and L. E. COPELAND
Authorized Reprint from
Highway Research Board Special Reporc
The Flow of Water in Hardened
Portland Cement Paste
By T. C. Powers, H. M. Mann
AND L. E. COPELAND
Portland Cement Association
Research and Development Laboratories
5420 Old Orchard Road
Flow of Water in Hardened
Portland Cement Paste
TC POWERS, H.M.MANN, and L. E. COPELAND,
Research and Development Division, Portland Cement Association
Permeability ol hydrated Portland cement pastes w" ">easured at to ur .
temperatures. A theory based on viscous drag of fluid onpartc s
takine into account the effect of adsorption on viscosity, was applied.
Partfcle size w^s estimated as *" 5 = 194A or 20lA, first and second
r P proximaaonI where + is the sphericity factor and 8 the volume
mTmeTer The specific surface diameter, obtained from specific
furTace of the soUd phase as measured by water vapor adsorption was
109A or 95A This gave + = 0. 46 or 0. 36, indicating sheets or fibers,
in agreement with electron-micrographs obtained in l»iw.
»<5TnnTFq of the oermeability of hardened cement paste to liquid water were carried
!„ S a™ ItlraT^ans'of 'studying the colloidal structure of hydrated cement paste
carman's (1) successful adaptation of the Kozeny (2) analysis of fluid flow in g^anuiar
Si JcXdUe possibility of the same Rind of ^cat^^^^«
was known to be composed principally of colloidal particles. First ; ltt « I "P t ^° e a PP > d
The Carman-Kozeny approach were encouraging, but certain ^repancie f s »ere "O^Kf
The discrepancies between theory and experimental results proved o £ fun dan^ntany
significant, and finally the Carman-Kozeny approach was set aside •» favor of an anal
ysfs based on the Stokes concept ot viscous drag as developed by S™. •(?). Alter
modification as required by the extreme smallness of the water -filled spaces ^
dened cement paste, this analysis led to a measure of ■ parUcle size terms')! sort a
diameter," and "sphericity factor." This value agreed wlt ^ e a r n ^°^°"; e P I . ar
larly specific surface as determined by water vapor adsorption and the Br " nauer
Emmett- Teller equation (4) and electron-micrographs obtained '"l 9 ".^ present
paper describes the experimental methods and results, gives the theoretical basis
analysis, and shows the degree of fit between theory and experiment.
METHODS AND RESULTS
Preparation of Test Specimens
This paper is based on many experiments carried out over * Pe r f d oI abou | ie "
vears Although precision oi measurement was adequate, data obtained were generally
difficult to deaf wfth because of differences between flow rates or supposedly ident 1
samples. Early attempts at theoretical analysis were only P"^^ 6 ^^.
the data points were too scattered to settle crucial questions. Finally, i .was ■**
covered ihat the principal cause of variations among similar samples « ma lid* Her
ences in alkali content ol the solution in the pores of the specimens. Onl th * _fuul «
periments were used directly in this paper. They were made on specimens preparea
lO J LOWS * A
A group of specimens was prepared from which nearly all the ^l^' e ^f Ted
by allowing it to diffuse into the curing water. Conditions were such that this required
more than 600 days. The permeability coefficients were measured from time to time
during the 600-day period, on companion samples. Gradual removal of a ^U was
shown by a slow increase in coefficient of permeability, the final coeffic en exceeding
the lowest bv a factor of 5 or 6. and by chemical analysis at the end of the test. < ver
beck, in unpublished work, had previously shown that soluble materials reduced rate o
flow through specimens of paste much more than could be accounted for by the ordinary
effects of solutes on viscosity of water. ) .
These special specimens were made from an ultra-fine cement to obtain a nnai
product practically free from unhydrated cement, and so that a wide range oi water-
cement ratios could be used without appreciable bleeding during the plastic stage. The
special cement was prepared by passing 30 lb of a commercial type I product through
a small air separator and discarding the coarser two-thirds. The resulting ultra-fine
product had a specific surface of about 8,000 g per sq cm as measured by the Blaine
meter. The hydration products from the ultra-fine cement showed about the same phy-
sical characteristics as those from a cement of ordinary fineness prepared from the
Neat cement pastes were prepared and cast in glass tubes in the way described in a
previous publication (5). Four different ratios of water to cement were used: 0. 5, 0.6,
0. 7 and 0. 8. Specimens having still higher water cement ratios were prepared and
tested, but results are not reported here because not all the alkali was removed from
them before and during the test.
T T I ■ it 1
— l —
No 1, K, * 42.8 a 10"'* cm/sec.
No 2, K," 32.2
No 3, K,= 219
No. 4, K,= 15 2
No 5, K,» 48 2
40 80 120 160 200 240
Applied hydraulic head, Ah, cm Hg
40 80 120 160 200
Applied hydraulic head, Ah, cm Hg
Figure 1. Rate of flow vs hydraulic head
— W/C =0-5, Specimen 4-1^-B.
Figure 2. Rate of flow vs hydraulic head
— W/C =0.5, Specimen k-±k -A.
No I,K,*I2I »I0
No 2,K,--85 I
No 4, K =44.2
No. 5,K =1260
40 80 120 160 200
Applied hydraulic head, Ah, cm Hg
No . 1, K,= 317*10" cm /sec
No, 2, K,*230
No. 4. K,« 127
No. 5. K,= 361
40 80 120 160 200 240
Applied hydraulic head, Ah ? cm. Hg
Figure 3. Rate of flow vs hydraulic head
— W/C =0.7, Specimen 14-20 C.
Figure 4. Rate of flow vs hydraulic head
— W/C =0.8, Specimen 1-2&C.
The molds were not quite filled with paste, and immediately after placing the paste
in the mold, 15 to 20 cc of water was added, and the mold was stoppered. The mold
was then stored in a constant-temperature room (23 C) for about 20 months. During
this time most of the alkali in the specimens diffused into the water at the top of the
specimen (the "curing water"). Practically all the rest of it was removed while the
specimen was under test, and before final rates of flow were established.
The mold was removed from the hardened cylinder of paste and one or more trun-
cated cones for permeability tests were obtained by means of a lathe and a diamond
saw, as previously described (5). After the permeability test was completed, each
test' specimen was analyzed for non-e vapor able water content, evaporable water content,
solid content, and alkali content. Cement and water were mixed under reduced pres-
sure to obtain paste without air bubbles.
Coefficients of permeability were determined by means of the apparatus described
before (5), with a modification. The modification consisted of equipping each of the
two water baths with a cooling coil. This made it possible to measure the coefficient
of permeability of each specimen at four different temperatures. At each temperature,
rates of flow were measured at three different pressures, approximately 1, l/ 2 and 3
atmospheres above ambient pressure. Sixty rates of flow were established, four at a
time; two to five days being required to establish a rate at a given pressure.
The relationships between applied pressure and measured rate of flow are shown in
Figures 1, 2, 3 and 4. Where two or three points for a given pressure are shown, they
represent rates observed on successive days. A plot of such readings vs time shows
that the differences are partly due to inaccuracy in measuring the exceedingly low rates
of flow and partly due to small changes toward the final steady rate. The positions of
the straight lines were established usually by the average of the last two or three read-
ings at each pressure. Although some deviations may be seen, in general an increase
in applied pressure produced a proportionate increase in rate of flow.
In most cases the best straight line for a given specimen and temperature does not
pass through the origin. This is due to osmotic pressure, which in some cases aids
the applied pressure and in other cases opposes it. During the course of an experimen
it may vary in magnitude, and even may reverse direction, before final rates of flow
are established. The development of osmotic pressure may be ascribed to electrolytes
in the paste. The electrolytes tend to diffuse out of the specimen during a test and,
because the volume of fluid exposed to the upstream face per unit of specimen area is
not the same as that at the < vnstream face, and because of flow itself, small ttiffei
ences develop in concentrations of electrolyte. The test specimen functions as a 6 ai-
permeable membrane; thus osmotic pressure may be manifested. The principal elec-
trolytes are sodium hydroxide, potassium hydroxide, and calcium hydroxide.
The specimens dealt with here were so nearly free from alkali that osmotic pres-
sures were small, and in some cases nil. After the permeability tests were completed
the specimens were tested for alkali content by means of the flame photometer (ASTM
Designation CI 14). In each case the NaaO content was too low to be detected and the
K2O was found to be in order of ascending water cement ratios, 0. 003, 0. 003, 0. 005
and 0. 00 percent of cement weight. Osmotic pressure tends to change in the course of
an experiment, but for this group of specimens the change was so slow under final con-
ditions that the three flow-rates for a given temperature could be established before an
appreciable change in osmotic pressure occurred.
The coefficient of permeability is directly proportional to the slope of a line such as
one of those in Figure 1. The coefficient corresponding to each point in Figure 1 is
tabulated in Table 1 along with other data to be discussed later. The method of calcu-
lation was the same as described before (5).
DISCUSSION OF DATA
Hardened cement paste is predominantly a gel; that is, an aggregation of colloidal I
particles. It has high capacity for evaporable water, upwards of 30 percent by volume 1
OBSERVED COEFFICIENTS OF PERMEABILITY
^ solid content of spe< mien, cc per cc
ChronolOL J Temp Temp..
Order 8 Abs
(deg C) T
Permed bili t
Spt en 13t,-B
= 0. 586; -r-5—
1 - c
I4 a -A
, j _ c
Specimen 20a-C; c = 0.471; y^-y = 0.890
Spen 28 b -C; 0.428; j^- =0.748
as compared with 1 percent, more or less,
for typical rocks. Yet the coefficient of
permeability of paste is as low as that of
most rocks (5). This is a manifestation of
the paste's fineness of texture; the pores
and particles are exceedingly small and
numerous. A considerable fraction of, if
not all, the water contained in these pores
is adsorbed; that is, it is within the range
of mutual attraction between water mole-
cules and the solid surface. The question
arises, therefore, as to whether all the
water is mobile. Adsorbed water might
have ice-like structure, and might show a
"yield point," so that, at some threshold
pressure, water immobile at a given pres-
sure becomes mobile. If some of the water
is immobile at a given pressure it is likely
to be that in the first adsorbed layer, or
that held in the densest portions of the
structure, and in the wedge-shaped spaces
where fibers cross each other. So far as
flow is concerned, such immobile water
would be part of the solid phase. All the
particles would function as if they were larger than they are, and the total porosity
would seem smaller than the total space occupied by evaporable water. There is also
the possibility that water may enter the lattice of layered crystals, but not take part in
Indications of the data to be presented hereafter are that flow through cement paste
is not fundamentally different from flow through other porous bodies, even though flow
is undoubtedly influenced by adsorption. As already mentioned, the data plotted in
Figure 1 show conformity to Darcy's law. Many other data not included here confirm
this aspect of the results. It may be concluded from that observation that if any of the
water was immobile, it was immobile at all the pressures used in the experiments.
Further analysis developed later herein leads to the conclusion that none of the water
is immobile. There is no evidence that particles giving rise to viscous drag in the per-
meability test are larger than the particles of the solid phase; that is, there is no evi-
dence supporting the supposition previously stated that some of the water held in the
densest portions of the structure is immobile.
THEORY OF PERMEABILITY
Previous investigators have developed theories of flow through porous materials and
granular beds. The subject has been dealt with mainly from two points of view. One
view leads to considering the conduits in a granular bed to be analogous to a bundle of
parallel capillaries, and applying the Poiseuille-Hagen law of capillary flow, using hy-
draulic radius (volume of water-filled space divided by the wetted boundary area of that
space) as a measure of size of capillary. The Kozeny equation as used by Carman is
the principal expression of this idea. The other point of view leads to considering the
viscous drag of moving fluid on a particle. The drag may be developed by a particle
falling through a fluid, or by flow through a granular bed where the particles are in
fixed positions. This approach to the problem leads to a more general law than does
The theory of permeability based on the concept of viscous drag can be approached
by considering the drag on a single particle moving under gravitation through a large
body of fluid. The driving force is the net weight of the particle multiplied by the gra-
F = (p s - Pf) (
in which F = driving force;
p s = density of the solid material;
pf = density of the fluid;
g = gravitational constant (= 980 cm per sec per sec); and
8 = diameter of a sphere having the same volume as the actual particle.
The drag is a function of size and shape of particle, of velocity, and of viscosity of the
fluid; that is (Stokes),
R = 3ini(e)Vd d (2)
in which R = resistance (drag);
•q(9) = viscosity of fluid at 6 C, in poises;
V = velocity of fall, in cm per sec; and
dd = drag diameter (that is, the diameter of a sphere having the same viscous
drag as the actual particle) (6).
When force and drag are equal, the rate of fall is constant at "Stokes velocity," or
Vc= Hs^9iM_ (3 )
s dd '
v _ (Ps -Pf)gd s (A\
Vs - 18x1(9) (4)
which is "Stokes velocity 11 in terms of M Stokes diameter"; that is, Stokes Law for the
fall of an isolated particle.
When there are many equal particles suspended in a fluid medium, the fall of the
suspension is slower than that of an isolated particle of the same kind, and the rate is
lower the higher the concentration of particles. Thus,
V(c,9) = V s 4> (c) (5)
in which V(c, 8) is the rate of fail of the suspension at temperature, 0, in which the
volume concentration of particles is c, and in which the particles are characterized
by Stokes velocity, V s ; <j>o(c) is some function of the total volume of particles. The
latter function operates as a reduction factor.
From theoretical and empirical considerations, Steinour (3) found, for isothermal
4> S (c) = (1 - c) 2 exp(-ktc) (6)
where ki is a constant. (For spheres of tapioca and for microscopic glass spheres, he
found k x = 4. 19. )
From theoretical deductions by Vand (7), Hawksley (8) found
+H(c) = (1 - c) 1 e*p - (y%^) (7)
Here k2 is the same as the constant in Einstein's (9) equation for the effect of rigid
particles in a fluid on the apparent viscosity of the~fluid. Its theoretical value, 2. 5 for
spheres, has been confirmed experimentally for low values of c. Q is Vand's "hydro-
dynamic interaction" constant. Its theoretical value for spheres is about 0.6.
Moone (10) also arrived at Eq. 7 with the same value for k 2 , but with a different
interpretation and evaluation of Q. From his point of view, Q is a "crowding factor"
and depends on the range of particle sizes, if the system is polydisperse.
Eqs. 6 and 7 were compared by applying them to Steinour 1 s data for tapioca sus-
pended in oil. These data included a well-established rate of fall for an isolated parti-
cle, and values of c ranging up to about 0. 5. Three assumptions were made:
4>s( c ) = (1 " c ) 2 ex P ( - kic) (Steinour)
4>H(c) = (1 - c) exp t _ Q 6c (Hawksley)
o ~ ICO
4>M( C ) = (! - c) exp t c (Mooney)
These expressions were used in place of 4>o(c) in Eq. 5 and the constants not arbitrarily
fixed were evaluated from the data by means of an electronic computer. The results
For $ s (c), V s = 0. 120 ± 0. 003; ki = 4. 25 ± 0. 08.
For4> H (c), V s = 0.102 1 0.002; k = 2. 87 ±0.05; Q = 0. 6 (arbitrary).
For <j> M (c), V s = 0.110 1 0.004; k = 3. 51 ± 0. 27; Q = 0.31 ±0.11.
The average of 150 tests of rate of fall of a single particle in a 1,000-ml oil-filled
cylinder gave a velocity of 0. 1120 ± 0. 009 cm per sec. Correcting for wall effect in-
creased this figure to 0. 1194 t 0. 0095 cm per sec. Steinour 1 s function apparently
gives the most accurate prediction of the Stokes velocity.
Although the other functions do not fit the data quite as well as Steinour* s, they
might seem preferable because they include explicitly factors called for by analysis
based on hydrodynamic considerations. However, they apparently are not valid ex-
pressions of theory, for not only do they give a relatively inaccurate prediction of
Stokes velocity, but also they give incorrect values of k. The Einstein constant should
have come out as 2. 5 or perhaps slightly higher, because the tapioca particles were
spherical, but it did not do so for either <|>h(c) or <J>m( c )- ° n me whole, it seems that
Steinour' s simpler function takes into account adequately what the more explicit func-
tions failed to do. As will be seen later, a still different function is required for the
present case. At this point, the function will be left unspecified, except for the quan-
tity (1 - c) 2 ; all agree on that, because it arises from fairly obvious considerations of
the influence of presence of particles in the suspension on the buoyancy of a given par-
ticle, and on the upward velocity of water relative to a given particle. Hence, it may
be assumed that
<|>o(c) = (1 - c) 2 exp - 4>c (8)
for the specific case to which the ensuing discussion pertains.
The fall of a thick suspension of equal particles (monodisperse) or of a flocculated
suspension of unequal particles (polydisperse) leaves an accumulation of clear fluid
above the suspension. The velocity of fall, in cm per sec, is equal to the rate at which
the clear fluid appears in the container above the suspension. Thus, using Eqs. 3 and
5, it is found that
in which q is the efflux from the suspension, in cc; t, is time, in seconds; and A is the
area of the suspension, in sq cm.
By Darcy' s law
dq 1 KiAh
dt A L
in which Ki is the coefficient of permeability, in cm per sec, and -j- is the hydraulic
gradient in terms of hydraulic head.
In the suspension, the excess hydraulic gradient is due to the weight of the solids,
minus the weight of the fluid they displace; that is,
df^ =(Ps- Pf)c (11)
Kl = 144^ li^ll eX p(- $(c)) (12)
If Eq 12 represents a thick polydisperse suspension in which all particles fall at
the same rate, regardless of size, Stokes diameter, d s , represents a "typical particle"
which is not easy to define. In the present case, Stokes diameter will be replaced by
a "surface diameter," as will be seen further on.
Hawksley pointed out that in using a formulation such as Eq. 12, the streamlines
around each particle are assumed to be symmetrical with the general direction of flow,
as in the fall of an isolated particle. In a thick, flocculated suspension, or in a gran-
ular bed, where the individual particles are not free to adjust themselves so as to
equalize couples of forces acting on them, the patterns of flow deviate from the general
direction. Citing arguments of Fowler and Hertel (11), he inserted an "orientation
factor," £. which would appear in the first right-hand term of Eq. 12. This factor is
essentially the "tortuosity factor" required by the Kozeny-Carman approach, which is
practically constant (at least its product with the "shape factor" is constant)- for a con-
siderable range of c. Nevertheless, it is a function of c such that as c diminishes
lim £(c) = 1.0
For flocculated thick suspensions and for granular beds, £(c) may be assumed to be
%. In terms of the foregoing development this means that if the streamlines around
an isolated particle could be caused to deviate from the general direction of fall to the
same extent that the streamlines in a bed of particles deviate from the general direction
of flow, the average rate of fall of the isolated particle would be about % V s . Thus,
Eq. 12 should be written
K -^M i(e, <i^li exp(-*(c,) (13)
in which £(c) = % for thick flocculated suspensions or granular beds. Using Steinour's
data for uniform emery particles for both the dispersed and flocculated states, Hawksley
(8) found I - 0. 71 for the flocculated state, which is considered satisfactory agreement
with the theoretical value, % •
APPLICATION TO HARDENED PASTE
Hardened paste, composed of hydrated portland cement, is a porous solid and not
an aggregation of discrete particles. Nevertheless, it is composed of particles, and
the connections between particles apparently involve but a small fraction of the total
surface of each particle. Therefore, to treat the material as an aggregation of dis-
crete particles involves no conceptual difficulty. The particles in hard paste are ex-
ceedingly small, but smallness per se would not seem to preclude application of the
theory as long as the particles are large relative to water molecules. A major diffi-
culty does arise, however, from extreme smallness of interparticle spaces. Because
of this, most of the fluid is in the force-field of the solid particles; that is, much oi
the water is adsorbed, as previously noted. Moreover, the force-field contains hy-
drated ions. The net effect is that the viscosity of the fluid cannot be treated as a con-
stant at a given temperature; it is a function of the dimensions of interparticle space,
and of the kinds and amounts of dissolved materials.
Adsorption forces probably cause the pattern oi flow to be different from laminar
flow in a uniform field. Nevertheless, experiment shows that for any given specimen
of paste, the rate of flow is proportional to the pressure gradient (see Figs. 1 to 4).
Therefore, in a given specimen, the coefficient of viscosity indicates the average rate
of shear under unit stress. In other words, a coefficient of viscosity found in a given
specimen of paste has the same significance that it ordinarily has. It differs from the
ordinary coefficient in that it is not a property of the fluid alone.
Because some of the water in saturated hydrated cement paste, perhaps all of it, is
subject to adsorption forces, viscosity should vary from point to point in the specimen,
and because the effects of adsorption are
various, the average viscosity could be a
function of the concentration of particles.
Accordingly, r\ (8) in Eq. 13 should be re-
placed by j\ (6)t| (c), the function t\ (c) being
a factor by which the normal viscosity is
multiplied to obtain the actual average
viscosity in the specimen. Making this
substitution and taking logarithms
InKi = InB + In
- lnii(e) -
lrn^(c) - 4>(c)
= A s for a granular solid
Differentiation of Eq. 14 with respect
to the reciprocal of temperature gives
Figure 5. Permeability as a function of
temperature: T = absolute temperature; K
= permeability coefficient, in cm per sec;
and - slope = 896 + 6^-7 ( ± C _ c ).
According to the theory of Eyring (12), this derivative of viscosity is a measure of the
activation energy for flow. The first of the two derivatives in brackets would be the
normal activation energy of the fluid; the second, the added activation energy required
when the fluid is adsorbed. For non-associated liquids, activation energy, hence the
first derivative on the right side, would be constant for a considerable range of tem-
perature. However, for water it varies about 10 percent from the average for the
range to 30 C. The empirical plots in Figure 5 show, however, that it is nearly con-
stant for water in paste; the variation of the derivative representing the effect of ad-
sorption is apparently equal and opposite to the variation of that for free water over the
temperature interval of these experiments. It is assumed, however, that the derivative
of the part due to adsorption is a constant at a given particle concentration.
The effect of particle concentration on activation energy should depend on average
distance between solid surfaces; here it is assumed to be proportional to the inverse of
hydraulic radius, which is proportional to y^
dlnii(c) m a(
1 - c
in which a is an empirical constant. Integrated, this gives
lnt|(c)=-| (yrr) + constant
Since lim r\(c) = 1. 0, the constant of integration is 0.
If it is assumed that 4>(c) of Eq. 14 is proportional to the inverse of hydraulic radius,
and is independent of temperature,
with v being an empirical constant.
Substituting from Eqs. 17 and 18 into
Eq. 14 gives
rT ^ ) * =lnB-lm 1 (e)-(^ + 7) I T
This semi-empirical equation indicates
that for data obtained at a given tempera-
ture, experimental values of the member
on the left plotted against .
produce straight lines, one for each tem-
perature. Trial plots of the data in Table
1 (not reproduced here) show that the
points conform very well to the expected
Fit of Data
COMPARISON OF OBSERVED
VALUES OF Ki AND
FROM EQ. 20
1.36 x 10T* § a =
1242 and y = 0.7
The fit of the data in Table 1 to Eq. 19
was determined by the method of least
squares. The first item of each group of
four was omitted from calculations be-
cause of the following considerations:
The five points of each set were obtained in the sequence indicated in the first column.
The first flow rate was established at 27 C, the second, third, and fourth at succes-
sively lower temperatures, and the fifth at the original temperature. The first test at
27 C gave a lower value of Ki than did the last. This change was probably caused by
loss of alkali from the specimen during the first test. The indications are that there
was little or no further loss of alkali during the four subsequent tests. From the other
16 values, the least square calculation, done by electronic computer, gave the follow-
ing values and standard deviations:
B = (1.36 ± 0. 1) 10" 10
a = 1242 + 133
y = 0. 7 - 0. 5
Using these values the following empirical equation for the coefficient of permeability
of the specimens represented in Table 1 is obtained:
1.36 x 10 w (1 - c 2
Ki = rprx — exp
24? + 0.7)^1
Values of Ki computed from Eq. 20 are compared with observed values in Table 2.
INDICATED PARTICLE SIZE
One object of this investigation was to determine what size of particle would be in-
dicated by measurement of fluid flow, and to compare the result with indications from
other kinds of measurement. Along with size there is also shape of particle to be con-
sidered. Studies by Ake Grudemo during 1957 showed, by means of electron micro-
scopy, that the particles in hardened paste have various shapes. The calcium silicate
hydrates are (a) ribbon-like fibers, somewhat rolled up edgewise, and (b) sheets or
foils aggregated as fluffy masses. The calcium aluminate hydrate and the calciumalu-
minoferrite hydrates occur as flat plates individually surrounded by calcium silicate
hydrate. Something of size and shape can be expressed in terms of appropriate "equi-
During 1957, Ake Grudemo, of the Swedish Cement and Concrete Institute, was a
guest scientist in the Portland Cement Association Laboratories, Skokie, 111. His
work is to be published.
valent spheres" and a "sphericity factor." Hawksley (6) summarizes the following hy-
pothetical sphere diameters, three of which were previously defined:
8 = volume diameter: diameter of sphere having the same volume as the particle.
d<i = drag diameter: diameter of sphere having the same viscous drag as the
particle in a fluid of the same viscosity and same velocity.
A = surface diameter: diameter of sphere having the same external surface area
as the particle.
d s = Stokes diameter: diameter of sphere having the same density and same free-
falling speed as the particle in a given fluid.
d = specific -surface diameter: diameter of a sphere having the same ratio of
external surface area to volume as the particle.
$ = sphericity factor: the ratio of surface area of a sphere having the same vol-
ume as the particle to the actual surface area of the particle.
d = <t> 5 (21)
d= 4>%d s (22)
Owing to the smallness of interparticle spaces and to the force field within such spaces,
streamlining of irregular particles probably does not occur. Viscous drag would seem
to depend, therefore, on the extent of surface presented by the solid phase, and the
average interparticle distance. The shape of the "typical particle" is probably not im-
portant in that respect. Accordingly, it is possible to identify the drag diameter with
the surface diameter under the conditions of flow presumed, and thus, with reference
to the relationship shown after Eq. 3:
6 3 8 3
From Eq. 23
_ = $ 8
The definition of B given with Eq. 14 thus becomes
= Pgfr 2 S 2 (24)
Pf = 1.0 (assumed) and g = 980 cm per sec per sec.
Using the value of B given above,
A^i.sexio- 10 =3>75xl0 -*
4> /4 8 = 194A
From definitions previously given, the specific -surface diameter is 4>8. Hence, to
evaluate <j> the specific -surface diameter is needed.
In 1947, Powers and Brownyard (15) gave the specific -surface diameter of the par-
ticles in hardened cement paste as 140 A. This value has been revised downward be-
cause of the following considerations:
According to the BET theory,
N Vm (25)
<t = specific surface area by BET method, in cm" ;
N = Avagadro's number, 6.023 x 10 23 ;
M = molecular weight of water, 18 g per mol;
V m = water required to cover the surface with a monomolecular layer, g; and
a = area covered by 1 molecule, in sq cm.
Powers and Brownyard let a = 10. 6 A 2 per molecule, and their equation was
a =(35.7xl0 6 )Yj]l (26)
More recently, Brunauer (16) showed that a = 11.4 A 2 is a better value for these sys-
tems than is 10. 6. On this basis the numerical coefficient becomes 38. 5x10, which
gives a specific -surface diameter about 7 percent smaller than that given by the first
As described by Copeland and Hayes (14) in 1953, the method of drying samples of
paste preparatory to adsorption measurement has been changed since the work reported
by Powers and Brownyard was done, so that the non-evaporabie water content is about
8 pen it less than it was when determined by the original method. This change also
increased V m by about 13 percent; therefore, a present-day estimate of specific sur-
face diameter from V m would be further reduced. (There is reason to believe that the
present method of drying gives more nearly the desired result than did the old. ) A
given sa pie treated by the 1953 method would give a higher value of cr than it would by
the 1947 method, as follows:
<T53 38. 5 _ t ,., CI47
3577 - 113 =*
Hence if d, 7 -- 140 .. d = 114 A
T i a partial correction of the 1947 et ate. A further correction is based on
the amount of unhydrated cement ie sampl< Tin 'typical paste" for which Powers
and Brownyard calculated d = 140 A probably contained out 5 percent of unhydrated
ment in th< orm of relatively large particles of clinker. Therefore, the estimate,
lsed not i new value of a and new method of drying (new value of Vm ), but
a. > for unhydrated cement, would be
d = 114 x 0.95 -- 108 A
which i the >inal< rre< d value corresponding to thi or nal 140 A.
or th' -;• lmens r« res< d in 1 tie 1. < mations of V m were made by the
1953 i hod ttu amples c ed no unhydrated i ment, and th i c< lent was chem
ically th ame as that to wh h I Powei nd Brownyard vaiu< rtained. There-
fore, the values obtained are direct 1. con tbf irith thi >ri d Pov sand
Br ird \ ue. The r ult.^ are \ Ln Table 3.
The data idicate that
<r= (38.5 x 10 6 ) 0.143 = 5.5 x 10' <m~ l
d = - = )9
is i good agreement with th<
alu> : Powers and B at. rd.
Data in T. W that ( hie of
—* a , and h« I I specific surfa >l
ne soli' s the same in each ol the four
specin is regardless of dif: ices in
1 - c
13b " B
14 a - A
20 a - C
2 - C
density. This means that the specific TABLE 4
surface of the hydration products is in- SPHERICITY FACTORS FOR VARIOUS
dependent of the original water/cement PARTICLES
ratio of the paste. Various other data not
given here support this indication that the Kind q£ ^^ +
hydration product of a given cement, hy-
drated at standard temperature, has a Sphere 1. a
characteristic specific surface. Other Portland cement 0, 7 to 0.8
data show that specific surface varies with Emery 0. 68 to 0. 72
chemical composition of the cement, but Ground glass 0. 71 to 0. 82
the variation is small. Fusain fibers 0. 38
Mica flakes 0. 28
Sphericity Factor a By def inition .
If it is assumed that values of d and
(f* 1 ^ 6 obtained in the foregoing pertain to the same particles, the value of $ can be
computed. Since d = <|>5 (by definition) and <J> 1/4 6 = 194 A (by experiment), it follows
that <|> 3/4 = 109/194 = 0. 56, and <|> = 0. 46. This value for sphericity factor may be
compared with values in Table 4, cited by Hawksley (6).
The value <t> = 0. 46 may be considered a first approximation. A second approxima-
tion may be obtained by considering the effect of calcium hydroxide crystals. Besides
gel, the specimens contained crystalline calcium hydroxide, the crystals being much
larger than gel particles and thus having a relatively low specific surface. Studies of
related systems by Brunauer (17) indicate that the specific surface is not greater than
0. 5 x 10 6 cm" 1 . If is is assumed that erg = specific surface of gel particles, n x = their
volume, and n 2 = the volume of calcium hydroxide,
ni<r R + n(0.5xlO ) = a = 5 5 x 10
ni + na
The weight of total solids in these specimens was about 1. 234 g per g of original ce
ment in the specimen, and the weight of calcium hydroxide was about 0. 163 grams.
ni + n2 = —
ni = -
n 2 = -
W = original weight of cement in the specimen; and
Ps, pi, p2 = density of total solids, gel, and calcium hydroxide, respectively.
For the present purpose, it will suffice to assume that pi = p 2 = p s , which is not far
from the truth. Then
m 1-071 = Q>868
ni + n 2 1. 234
112 = £41! =0.132
ni + n 2 1. 234
5. 5 x 10 6 - 0. 6 66 x 10° - 9R v 10 e
a g = 07868 = 6. 26 x 10 ,
say cr g = 6. 3 x 10 6 cm" 1 . This gives a revised specific -surface diameter of d = 95 A.
It is now necessary to calculate a corresponding correction to the particle size in-
dicated by the permeability measurement. Because crystals of calcium hydroxide are
much larger than gel particles, they may be thought of as impermeable bodies embedded
in gel reducing the fractional area through which flow may occur. If for the present
purpose gel is defined as the porous substance occupying space not occupied by calcium
hydroxide crystals, it follows that for a given gel (so defined) the coefficient of per-
meability is proportional to the gel content. In a hypothetical specimen containing no
crystals of calcium hydroxide, the gel content is unity. In a real specimen the coeffi-
cient is smaller; that is,
ni + ri2
1 - 0. 132 c
where K e is the coefficient of permeability of pure gel. It is apparent that the magni-
tude of the correction depends on c and is therefore different for each specimen. It
will suffice to use the average c for the four specimens, which is 0. 50. This value
Kg = 1.07 Ki
Referring now to B as defined by Eqs. 14 and 15, B g will be considered that parameter
for pure gel. Then, with c constant,
= ±^g! = 1.07
hence <|> /4 = ^ = 0. 47 and <J> = 0. 36.
Indicated Size and Shape
On the basis of these values arrived,
paste may be described as follows:
the "typical particle" in hydrated cement
Sphericity factor, <fy
Spec. surf. dia. , d
Volume diameter, 4>
Surface diameter, A
Stokes diameter, d s
These values and Grudemo' s electron- micrographs agree in that each indicates the
particles in cement gel to be not spherical. From the electron- micrographs already
mentioned, and electron diffraction patterns, it appears that paste is composed mostly
of ribbon-like fibers, plates, and crumpled sheets, the substance being mostly amor-
phous or poorly crystallized. A value for specific surface estimated from measure-
ments of micrographs indicates (but does not prove) that these are the very particles
that adsorb water vapor.
The foregoing values rest on the assumption that water-vapor adsorption measures
the same surfaces that are presented to flowing water. To test this assumption let it
now be assumed that the particles are layered structures such that water vapor can
penetrate and measure two "inside" surfaces for every two outside surfaces (edge
areas can be neglected). If it is assumed that the outside surfaces are accessible to
flowing water whereas the inside surfaces are not, the specific -surface diameter, d,
of the particles as felt by flowing water is one-half that indicated by adsorption. On
this basis the calculation of <J> would be, using the second approximation of d,
= 0.98; <|)= 0.97
This result indicates that the particles felt by flowing water are spheres, which is
known to be false.
Let it now be assumed that interlayer water can measure only one internal surface;
that is, that only one layer of water molecules can be accommodated in the crystal
lattice. In this case the specific surface of the particles felt by flowing water would
be two-thirds that indicated by adsorption. The indicated sphericity factor would be
4> 3/4 = L i 9 4 95 =0.73; 4> = 0. 66
According to Table 4, such a sphericity factor is typical of shapes found in crushed
brittle materials. This result also is contrary to evidence from electron- micrographs.
Alternatively, it could be assumed that water vapor measures only the outer sur-
faces of gel particles, and that the particles felt by flowing water are aggregations of
those gel particles. This would mean approximately that the "permeability particle"
is some multiple of the "adsorption particle." Any such assumption leads to sphericity
factors exceeding unity if the number of particles per aggregation exceeds 2, and the
result is therefore useless.
Thus, it seems that the original calculation, based on the assumption that adsorp-
tion measures the same particles that produce drag in the permeability test, is the
only one that gives an acceptable result. Possibly, a paste contains some crystals so
well developed as to admit inter layer adsorption, and possibly water could not flow
through such crystals. However, these data indicate that if such crystals exist, and
if they do exclude flowing water, they constitute a minor faction of the solids in the
paste. If is seems absurd to suggest the possibility of flow through a crystal, it should
be recalled that the flow under consideration is exceedingly slow, as shown in Figure 1.
A typical average rate is 20 x 10~ 9 cm per sec, which is about % in. per year. Since
viscosity is not uniform, flow in some portions of the structure may be several orders
of magnitude slower than the average rate. There seems to be no reason to suppose
that physically adsorbed molecules would be immobile under a hydrostatic pressure
gradient, whatever their situation might be.
These data indicate that all the evaporable water (as defined by Copeland and Hayes
( 14 ) ) is mobile under a hydrostatic pressure gradient, although some of it has high
viscosity. If this is so, none of the evaporable water has a finite yield point under
stress. Any change of stress in a concrete member, however small that change may
be, should start a redistribution of evaporable water in the member, and such redis-
tribution should be accompanied by shrinkage where the water content diminishes and
swelling where it increases. This has long been recognized as a factor characterizing
creep of concrete under sustained stress. It may be inferred that as far as this aspect
of creep is concerned "seepage," the threshold stress for creep of concrete members,
is zero, as indeed it seems to be in some experiments.
Of the two kinds of pores in hardened paste, gel pores and capillaries, the capil-
laries are continuous in some specimens, but not continuous in others. When they are
not continuous, they are called capillary cavities, and are interconnected only by gel
pores. Absence of continuous capillaries in the specimens used for this study is im-
plied by the conclusion that adsorption measures the same surface area as that pre-
sented to flowing water. This conclusion is strengthened by data from other specimens
not presented here showing unmistakable evidence of continuous capillaries in those
specimens. This subject is to be dealt with in a separate paper.
If the volume diameter of a gel particle is about 260A, as indicated by these data,
the corresponding number of particles per unit volume of gel can be computed. The
solid content of a unit volume of gel (exclusive of capillary cavities) is about 0. 65.
Hence, the quotient of 0. 65 and volume per particle is the desired number. The vol-
ume per particle turns out to be 9 x 10 cc, or 0. 6 x 10" * cu in. Hence, the number
of particles is about 7 x 10 16 per cc, or 10 18 per cu in. of gel.
A major component of the gel is probably a tricalcium disilicate trihydrate having a
volume of about 140 cc per gram-molecular weight (16). This corresponds to about
2. 3 x 10~ 22 cc per molecule. A gel particle having a volume of 9 x 10" M cc would thus
contain about 40,000 molecules. This indicates the degree to which the water-solid
relationship in cement gel approaches (and misses) that of a homogeneous solid solution.
1 The flow of water through a given specimen of hydrated cement paste complies
with Darcy's law, but it is liable to be complicated by effects of osmotic pressure.
2 Within the range of paste porosity included in this report, part of the evaporable
water, perhaps all of it, is adsorbed. Consequently, viscosity is higher than that of
free water. (Also, viscosity is increased by the presence of solutes. )
3. Average viscosity of water in hydrated cement paste is a function of gel-particle
concentration and temperature.
4. Temperature dependence of rate of flow in paste indicates a relatively high ac-
tivation energy for flow, reflecting the effect of adsorption.
5. The theory of permeability based on the concept of viscous drag on particles as
developed by Steinour, but modified to allow for variable viscosity, and modified by
considerations advanced by Hawksley, gives a good fit to these data.
6. The particle size and shape indicated by permeability data, combined with ad-
sorption data, are as follows: (second approximation) sphericity factor, 0. 36; volume
diameter, 264A; surface diameter, 440A; Stokes diameter, 205A; specific surface
diameter. 95A. These data are compatible with electron- micrographs.
7. The sphericity factor obtained by combining permeability data with adsorption
data indicate the particles to be sheets, or fibers, or both, in agreement with electron-
8. All the evaporable water appears to be mobile.
9. The particles giving rise to viscous drag in the permeability test are apparently
the same particles, which, when dry, adsorb water vapor.
10. In the specimens studied, there apparently were no continuous capillaries by-
passing clusters of gel particles.
1. Carman, P.C., "Flow oi Gases Through Porous Media." Academic Press,
New York (1956).
2. Kozeny, J. S. B. , "Capillary Motion of Water in Soils" (Uber Kapillare des
Wassers Leitung in Boden). Akad. Wiss. Wien, Berichte 136, 2a (5-6): 271-306 (1927).
3. Steinour, H. H. , "Rate of Sedimentation: Nonflocculated Suspensions of Uniform
Spheres." Ind. and Eng. Chem. , 3.6:618-624(1944). "Suspensions of Uniform Angular
Particles." Ind. and Eng. Chem. , 36:840-847 (1944). "Concentrated Flocculated
Suspensions of Powders. " Ind. and Eng. Chem., 36:901-907(1944). All in Portland
Cement Assn. Research Dept. Bull. 3.
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sity Press (1943).
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of Portland Cement Paste." Proc. ACI, 51: 285 (1954). Portland Cement Assn. , Re-
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Dynamics and the Stokes Diameter." British Coal Util. Research Assn. , Bull. 25: 4,
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and Flow Through Porous Media." In "Some Aspects of Fluid Flow," pp. 114-135.
Edward Arnold and Co. . London (1951).
9. Einstein. Albert, "A New Determination of Mol ular Dimensions" (Ein Neue
Bestimmung der Molecular Dimensioned). Ann. der Phy. , Leipzig, 19:289(1906).
10 Mooney, M. . "The Viscosity of a Concentrated Suspension of Spherical Particles
Jour. Colloid Sci. . 6: 162-170 (1951).
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Jour. Appl. Physics, 1 496 (1940).
12. Eyring, H. , "Viscosity, Plasticity and Diffusion as Examples of Absolute Re-
action Rates." Jour. Chem. Phys. , 4:283(1936).
13. Powers, T. C. , and Brownyard, T. L. , "Studies of the Physical Properties of
Hardened Portland Cement Paste, Part 2, Theoretical Interpretation of Adsorption
Data." (see p. 498) Proc. ACI, 43: 469-504 (1946); also, Portland Cement Assn. Bull.
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Water in Hardened Portland Cement Paste." ASTM Bull. 194 (Dec. 1953); also, Port-
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Hardened Portland Cement Paste." Proc. ACI, 43:270, Fig. 2-2 (1946); also, Portland
Cement Assn. Research Dept. Bull. 22, Part 2, p. 270.
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Dicalcium Silicate and Tricalcium Silicate at Room Temperature." Jour. Amer. Chem.
Soc, 80:761 (1958); Portland Cement Assn. Research Bull. 86.